LEAST SQUARES ADJUSTMENTS OF TRIANGULATION: DIRECTIONS VERSUS ANGLES

Abstract

This paper continues the discussion started in an article of the same title (E.S.R., ×, 78, :353-66), on which a further letter was written in October, 1952. The amount of computation required originally was very considerable, and it was obviously impossible to publish it all. The recent letter was necessary to answer the suggestion that the agreement between errors put in and corrections obtained from the L.S. solution was not very close. It seemed sufficient to give the list of errors and corrections, leaving readers to judge for themselves. The correlation coefficient from the two sets of figures was 0.78, which looked quiteg90d. Unfortunately, it was not realized before that corrections from a L.S. solution cannot, legitimately, be compared with errors put in on directions unless a station correction is first applied to the errors to make the sum of the errors at each station equal to zero. This is one of the points about the direction method of adjustment which is not very easy to understand.     

THE EFFECT OF ALTITUDE ON TRAVERSE ERRORS

Abstract

I. These notes are the results of following up in some detail the well-known fact that the horizontal distance between two points at altitude h is greater, by an amount proportional to h, than the distance between the corresponding points at sea-level. Traverses based on rectangular coordinates are considered, with special reference to the residual errors left after adjusting the misclosures of such traverses without first eliminating errors due to altitude.     

基准站观测数据加密方法及其在差分GNSS后处理中的应用

当基准站采样率低于流动站时,不能用常规差分全球卫星导航系统（Global Navigation Satellite System,GNSS）后处理方法得到流动站所有历元坐标。针对此问题,给出了基于精密单点定位（precise point positioning,PPP）模型构造基准站非采样点上虚拟观测值的方法。该方法将接收机钟差、对流层天顶湿延迟从观测值误差中分离出来,同消电离层模糊度一起进行估计,利用基准站真实坐标获得卫地距,在此基础上计算相邻两个观测历元的残差进而拟合历元间非采样点残差,与卫地距、各项估计误差一起生成虚拟观测值。该方法保持了虚拟观测值的误差特性,尤其是基准站与流动站间的共性误差。该方法仅加密基准站数据,对流动站没有影响。算例结果表明,基准站在采样间隔30 s范围内,使用该方法加密的虚拟观测值与真实值有较好的一致性,采样间隔分别为30 s、15 s、5 s的虚拟测码伪距和载波相位观测值偏差的标准中误差分别在0.2 m和1.2周左右、0.1 m和0.7周左右、0.05 m和0.2周左右;在30 s采样间隔情况下,按照该方法处理后仍能满足厘米级定位精度的要求。     

A generalisation of the least-squares method for the adjustment of free networks

Summary The system of normal equations for the adjustment of a free network is a singular one. Therefore, a number of coordinates has to be fixed according to the matrix. The mean square errors and the error ellipses of such an adjustment are dependent on this choice. This paper gives a simple, direct method for the adjustment of free networks, where no coordinates need to be fixed. This is done by minimizing not only the sum of the squares of the weighted errorsV ^T PV=minimun but also the Euclidean norm of the vectorX and of the covariance matrixQ X ^T X=minimum trace (Q)=minimum This last condition is crucial for geodetic problems of this type.     

Preliminary results of the establishment of a marine geodetic point in the Pacific Ocean

In November 1968, a marine geodetic control point was established in the Pacific Ocean at a water depth of6,200 feet. The control point (reference point) consists of three underwater acoustic transponders, two of which are powered with lead-acid batteries and the third with an underwater radioisotope power source “URIPS” with a10- to20- year life expectancy. Four independent measuring techniques (LORAC airborne line-crossing, satellite, ship inertial, and acoustic techniques) were used to measure and determine the coordinates of the control point. Preliminary analysis of the acoustic and airborne data indicates that high accuracies can be achieved in the establishment of geodetic reference points at sea. Geodetic adjustment by the method of variation of coordinates yielded a standard point error of±50 to±66 feet in determining the unknown ship station. The original location of the ship station as determined by shipboard navigation equipment was off by about1,600 feet. Paper previously published in the Proceedings of the Second Marine Geodesy Symposium of the Marine Technology Society.     

A regression model-based method for indoor positioning with compound location fingerprints

ABSTRACT

This paper proposed and evaluated an estimation method for indoor positioning. The method combines location fingerprinting and dead reckoning differently from the conventional combinations. It uses compound location fingerprints, which are composed of radio fingerprints at multiple points of time, that is, at multiple positions, and displacements between them estimated by dead reckoning. To avoid errors accumulated from dead reckoning, the method uses short-range dead reckoning. The method was evaluated using 16 Bluetooth beacons installed in a student room with the dimensions of 11 × 5 m with furniture inside. The Received Signal Strength Indicator (RSSI) values of the beacons were collected at 30 measuring points, which were points at the intersections on a 1 × 1 m grid with no obstacles. A compound location fingerprint is composed of RSSI vectors at two points and a displacement vector between them. Random Forests (RF) was used to build regression models to estimate positions from location fingerprints. The root mean square error of position estimation was 0.87 m using 16 Bluetooth beacons. This error is lower than that received with a single-point baseline model, where a feature vector is composed of only RSSI values at one location. The results suggest that the proposed method is effective for indoor positioning.     

A SIMPLIFIED ANALYTICAL METHOD FOR THE ADJUSTMENT OF AERIAL TRIANGULATION

Abstract

This method, which has been used in connection with Wild A5 and A6 stereoplotting machines, was developed in an endeavour to retain as many as possible of the advantages of the analytical method without incurring the corresponding disadvantage of excessively lengthy computation. It is capable of being quickly learnt by even an inexperienced computer, and adequate checks on the accuracy of the numerical work are easily applied where not automatically provided. It has the advantage over the graphical method formerly used, that no control point is accepted as more reliable than another, equal weight being given to all, and an indication of the relative reliability of the individual control points is furnished by the table of residual errors obtained.     

LEAST-SQUARE ADJUSTMENTS OF TRIANGULATION DIRECTIONS VERSUS ANGLES

Abstract

IT has been assumed in the past that because angles for triangulation are usually observed by the direction method, therefore it must be more correct theoretically to perform the least-square adjustments by directions rather than by angles. It is fairly obvious that an adjustment of the same figure by directions will not give the same result as an adjustment by angles: the unknowns in each case are different and the number of directions is usually about 25 per cent. greater than the number of angles for the same figure. Strictly, the least square method is only applicable to observations from which all systematic errors have been eliminated, and in which the remaining errors are truly accidental. It is generally safe to assume that most survey errOlS consist of a random and a systematic part. Rarely, if ever, is it possible to state that all systematic error has been eliminated, strive how we may to take all precautions against it.     

A bundle adjustment approach with inner constraints for the scaled orthographic projection

Bundle adjustment is a method for simultaneously calculating both the interior and exterior orientation parameters of a set of images, and the object-space coordinates of the observed points. In the case of long focal length lenses and narrow field-of-view (FOV) imaging situations, collinearity based (perspective projection) algorithms may result in linear dependencies between parameters that cause solution instability. The use of a scaled orthographic projection model based on linear algebraic formulations was therefore adopted to reduce this risk. Using quaternions, a new mathematical model is derived that includes the partial derivatives as well as the inner constraint equations for a scaled orthographic bundle adjustment. The model was then tested using two image sets of a single, small vessel (about 6 m length) with a cube target of known dimensions at two distinct ranges; perspective solutions were also calculated for comparison. RMS residual errors of 0.74-0.78 pixels associated with the new method compare favorably to a residual error range of 0.59-0.74 pixels using a perspective bundle adjustment of the same target points. Relative precisions (as a ratio of target size) of between 1:1650 and 1:750 have been achieved at ranges of 375 m and 850 m, respectively, given comparisons with the known cube dimensions. A third image dataset consisting of a network of 16 images was solved with a 1:2200 relative precision showing the new method can successfully handle high redundancy. For the experiments that were conducted, the new method was found to produce less precise results than the perspective bundle solution for a FOV of 0.50-0.65° where the object fills 5-8% of the image. However, it was found to match the precision of the perspective model (with an uncalibrated camera) for a FOV of 0.20-0.30° where the object of interest fills only 1-2% of the full image.     

A-10绝对重力仪残差谱分析

自由落体式绝对重力仪A-10的测量数据拟合残差呈现出明显的周期信号,对4个测站的残差进行FFT处理,并进行频谱分析,结果表明残差组平均的频谱更加稳定,主峰频率与测站无关,主峰幅值与测站明显相关,加周期项改正的绝对重力值偏小。     

Spatial triangulation in a local, astronomical oriented cartesian coordinate system

For the determination of spatial coordinates of the points of a net-work a local Cartesian coordinate system is introduced, whoze z-axis coincides with the true vertical of an arbitrarily selected central point in the net-work, and whose x-axis is oriented to astronomical north. The observed horizontal directions and vertical angles, which refer to the local verticals of the observation stations, are reduced to parallels to the z-axis and the x-axis. For this purpose the direction of the local vertical in the observation stations and at least the astronomical azimuth of one side of the net-work need to be measured. This method has the advantage, that the number of unknowns for one observation station is reduced from six for the conventional method to four, and that, if the vertical angle and the astronomical azimuth of a line are measured from both sides, a simple control of the measured values is possible before the adjustment. The formulae for the reduction of the measured quantities and the observation equations for the adjustment of reduced azimuths, reduced vertical angles and of distances are derived. The method is checked in a quadrilateral in the Swiss Alps and compared with the conventional ellipsoidal adjustment.     

自由设站法在点云拼接中的应用

为方便三维点云数据的统一管理和工程应用,需要为其提供统一的坐标基准。在数据采集阶段,对于在控制点可通视的测区内,可利用后视定向的方法完成点云扫描测量工作。针对控制点无法通视的测区,本文提出基于全站仪三维自由设站控制测量的方法,建立间接设站点平差模型并进行平差,进而解算标靶点坐标,利用标靶将扫描点转换到绝对坐标系中。试验结果表明:标靶坐标的测量精度和点云拼接的精度符合要求。在测区观测条件不好的情况下,自由设站法的应用为点云拼接提供了一种可行的方法。     

一种高精度的基坑水平位移监测方法探讨

将CPⅢ(高速铁路轨道控制网)平面控制测量的自由测站施测方法应用到基坑水平位移变形监测中,其特点是根据施工现场情况选择任意合适的地点架设测站,避免了施工环境对测量工作的影响和仪器的对中误差。介绍了Helmert方差分量估计定权原理和置平平差原理,通过自编程序完成了基坑变形监测的仿真计算。计算结果证明:利用自由测站采集观测数据,Helmert方差分量估计定权约束平差和置平平差均能够高精度地得到毫米级的基坑变形量,可以应用到基坑水平位移变形监测中。     

A unified approach to self-calibration of terrestrial laser scanners

In recent years, the method of self-calibration widely used in photogrammetry has been found suitable for the estimation of systematic errors in terrestrial laser scanners. Since high correlations can be present between the estimated parameters, ways to reduce them have to be found. This paper presents a unified approach to self-calibration of terrestrial laser scanners, where the parameters in a least-squares adjustment are treated as observations by assigning appropriate weights to them. The higher these weights are the lower the parameter correlations are expected to be. Self-calibration of a pulsed laser scanner Leica Scan Station was performed with the unified approach. The scanner position and orientation were determined during the measurements with the help of a total station, and the point clouds were directly georeferenced. The significant systematic errors were zero error in the laser rangefinder and vertical circle index error. Most parameter correlations were comparatively low. In part, precise knowledge of the horizontal coordinates of the scanner centre helped greatly to achieve low correlation between these parameters and the zero error. The approach was shown to be advantageous to the use of adjustment with stochastic (weighted) inner constraints where the parameter correlations were higher. At the same time, the collimation error could not be estimated reliably due to its high correlation with the scanner azimuth because of a limited vertical distribution of the targets in the calibration field. While this problem can be solved for a scanner with a nearly spherical field-of-view, it will complicate the calibration of scanners with limited vertical field-of-view. Investigations into the influence of precision of the scanner position and levelling on the adjustment results lead to two important findings. First, it is not necessary to level the scanner during the measurements when using the unified approach since the parameter correlations are relatively low anyway. Second, the scanner position has to be known with a precision of about 1 mm in order to get a reliable estimate of the zero error.     

THE ADJUSTMENT OF TRAVERSE NETS

Abstract

The weakest point in a straight traverse between two fixed points is well known to be in the middle. The uncertainty or p.e. perpendicular to the general direction of the traverse can be shown to be a maximum at the midpoint. Yet subsidiary traverses are usually tied in at or near this point, and consequently may show closing errors which are well in excess of what may be expected. A rigorous least squares solution would overcome this difficulty but only at the cost of a very laborious computation if the network is at all extensive. A compromise between rigour and labour can be achieved, however, which retains the major advantage of a fully rigorous solution, namely that the subsidiary traverses are not tied in at the weakest points of the main traverse system.     

地面站点叶面积指数观测的空间代表性评价——以CERN站网观测为例

在叶面积指数LAI(Leaf Area Index)产品真实性检验中,地面站点的多时相连续观测LAI数据是重要的验证数据来源。当站点观测范围与产品像元尺度不一致时,站点观测LAI直接用于产品验证可能为验证结果带来误差。因此,在验证之前需要分析站点观测对像元尺度的空间代表性,选择空间代表性好的观测来验证产品,从而减小尺度效应带来的验证误差。以往的研究只是简单的定性说明研究区域,并直接用站点测量数据对产品进行验证,缺少一套系统的站点观测在产品像元尺度内空间代表性评价的方法体系。本文提出了站点LAI观测的空间代表性评价方法,建立了评价指标DVTP(Dominant Vegetation Type Percent)、RSSE(Relative Spatial Sampling Error)和CS(Coefficient of Sill),构建了空间代表性评价分级体系。以中国生态系统研究网络CERN(Chinese Ecosystem Research Network)农田站和森林站LAI观测为例,对站点观测在1 km产品像元尺度内空间代表性进行评价,并分析评价前后站点观测对MODIS LAI产品验证精度的影响。结果显示,本文提出的方法能够有效地对不同站点LAI观测在产品像元尺度内空间代表性进行质量分级,且年际间的站点观测空间代表性较为一致。评价方法能够去掉在特定产品像元尺度下空间代表性不好的观测数据,一定程度上提高验证数据集对产品验证精度的可靠性。     

LOCAL ERRORS IN RELATION TO ANALYTICAL METHODS OF AIR TRIANGULATION

Abstract

In view of the limitations of the use of plotting machines in air triangulation a detailed investigation of the practical possibilities of the analytical methods has become necessary, especially now that inexpensive means of reducing the labour of calculation is in course of realisation.

In analytical methods it is practically possible to correct the measured photographic coordinates for determinable errors. Also, the use of a reseau as developed by the Ordnance Survey facilitates the reduction of several distortions. It is however argued and demonstrated that some errors of a local character remain without effective checking.

A discussion of the nature of local errors shows that the number of points usually employed for relative setting is too small to be effective in disclosing them. An increase of the number of observed points is therefore suggested, and should be expected to improve the accuracy in the meanwhile.

An assessment of the value of doubling the number of minor control points and of the improvement resulting from the use of reseau is made from a statistical analysis of data kindly provided by the Ordnance Survey.

Comparison of 89 overlaps carrying one centimetre reseau and 100 overlaps without the reseau shows that the use of reseau reduced the root mean square error of evaluation of want of correspondence by 13 per cent., which just passes the 10 per cent. point of significance. The use of double minor control on the reseau overlaps gave an improvement which is about equal to that due to the use of the reseau. Doubling the number of minor control points is shown to increase the effective number of degrees of freedom from one to an average of four, which should reduce effectively the chances of large errors passing unnoticed through the stages of relative setting.

Work on the frequency of local errors of various sources is in hand, and will probably be published in later issues of the Review.     

DISTORTING A NETWORK TO FIT CONTROL POINTS: AN ALTERNATIVE EXPRESSION FOR THE SCHOLS ADJUSTMENT

Abstract

Another form of Mr. Lauf's expression for a conformal adjustment of a system of coordinated points may be of interest. These are assumed to be already in harmony with i control points and are to be brought into agreement with j further points. (Mr. Lauf deals explicitly in his paper with the special case i = 2, j = 1, but he adumbrates a general solution.)     

CHECKING TRAVERSE COMPUTATIONS

Abstract

Traverse Computations must be Checked.—A traverse is a chain of points connected by angular and linear measurements. The check on observations is provided by the agreement, obtained in computations, between the terminals of the traverse (terminal bearings and terminal co-ordinates) taken as fixed. This check is not sufficient, however, to serve as a check on the computations. As a matter of principle, computations should be free of errors; there are no limits of tolerance in computational work except for discrepancies arising from inaccuracy of last figures. Secondly, errors in computation may occur that are not revealed by the traverse misclosures, not to speak of compensational errors, the field for which is very favourable in traverse work.